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B. Addition and Subtraction of Matrices

# B. Addition and Subtraction of Matrices

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Section 7.2 The Algebra of Matrices

EXAMPLE 4

653

Using Technology for Matrix Operations

Use a calculator to compute the difference A Ϫ B for the matrices given.

2

11

A ϭ £ 0.9

0

Solution

Ϫ0.5

3

4

6

6

5

Ϫ4 §

5

Ϫ12

Bϭ £

Ϫ7

10

1

6

11

25

0

Ϫ4

Ϫ5

9

0.75

Ϫ5 §

Ϫ5

12

The entries for matrix A are shown in Figure 7.9. After entering matrix B, exit to

the home screen [ 2nd MODE (QUIT)], call up matrix A, press the ؊ (subtract)

key, then call up matrix B and press . The calculator quickly finds the difference

and displays the results shown in Figure 7.10. The last line on the screen shows the

result can be stored for future use in a new matrix C by pressing the STO key,

calling up matrix C, and pressing .

ENTER

ENTER

Figure 7.9

Figure 7.10

Now try Exercises 21 through 24

Figure 7.11

In Figure 7.10 the dots to the right on the calculator screen indicate there are additional digits or matrix columns that can’t fit on the display, as often happens with larger

matrices or decimal numbers. Sometimes, converting entries to fraction form will provide a display that’s easier to read. Here, this is done by calling up the matrix C, and

using the MATH 1: ᮣ Frac option. After pressing , all entries are converted to fractions in simplest form (where possible), as in Figure 7.11. The third column can be

viewed by pressing the right arrow.

Since the addition of two matrices is defined as the sum of corresponding entries,

we find the properties of matrix addition closely resemble those of real number addition. Similar to standard algebraic properties, ϪA represents the product Ϫ1 # A and

any subtraction can be rewritten as an algebraic sum: A Ϫ B ϭ A ϩ 1ϪB2. As noted in

the properties box, for any matrix A, the sum A ϩ 1ϪA2 will yield the zero matrix Z,

a matrix of like size whose entries are all zeroes. Also note that matrix ϪA is the

ENTER

Given matrices A, B, C, and Z are m ϫ n matrices, with Z the zero matrix. Then,

B. You’ve just seen how

matrices

I. A ϩ B ϭ B ϩ A

¡

II. 1A ϩ B2 ϩ C ϭ A ϩ 1B ϩ C2 S

III. A ϩ Z ϭ Z ϩ A ϭ A

¡

IV. A ϩ 1ϪA2 ϭ 1ϪA2 ϩ A ϭ Z ¡

ϪA is the additive inverse of A

C. Matrices and Multiplication

The algebraic terms 2a and ab have counterparts in matrix algebra. The product 2A

represents a constant times a matrix and is called scalar multiplication. The product

AB represents the product of two matrices.

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CHAPTER 7 Matrices and Matrix Applications

Scalar Multiplication

Scalar multiplication is defined by taking the product of the constant with each entry

in the matrix, forming a new matrix of like size. In symbols, for any real number k and

matrix A, kA ϭ 3kaij 4.

EXAMPLE 5

Computing Operations on Matrices

4

3

3

1 § and B ϭ £ 0

Given A ϭ £

0 Ϫ3

Ϫ4

1

1

a. 2 B

b. Ϫ4A Ϫ 2 B

1

2

Solution

Ϫ2

6 § , compute the following:

0.4

3

Ϫ2

1

1

a. B ϭ a b £ 0

6 §

2

2

Ϫ4 0.4

3

1 12 2132

1 12 2 1Ϫ22

Ϫ1

2

1

1

ϭ £ 1 2 2102

1 2 2 162 § ϭ £ 0

3 §

Ϫ2 0.2

1 12 21Ϫ42 1 12 210.42

1

1

b. Ϫ4A Ϫ B ϭ Ϫ4A ϩ aϪ b B rewrite using algebraic addition

2

2

1Ϫ12 2 1Ϫ22

1Ϫ42142

1Ϫ42132

1Ϫ12 2132

ϭ £ 1Ϫ42 1 12 2

1Ϫ42112 § ϩ £ 1Ϫ12 2102

1Ϫ12 2162 §

1

1Ϫ42102 1Ϫ421Ϫ32

1Ϫ2 21Ϫ42 1Ϫ12 210.42

Ϫ16 Ϫ12

1

Ϫ32

ϭ £ Ϫ2

Ϫ4 § ϩ £ 0

Ϫ3 § simplify

0

12

2

Ϫ0.2

Ϫ16 ϩ 1Ϫ32 2

Ϫ12 ϩ 1

Ϫ11

Ϫ35

2

ϭ £ Ϫ2 ϩ 0

Ϫ4 ϩ 1Ϫ32 § ϭ £ Ϫ2

Ϫ7 § result

0ϩ2

12 ϩ 1Ϫ0.22

2

11.8

Now try Exercises 25 through 28

Matrix Multiplication

Consider a cable company offering three different levels of Internet service: Bronze—

fast, Silver—very fast, and Gold—lightning fast. Table 7.1 shows the number and

types of programs sold to households and businesses for the week. Each program has

an incentive package consisting of a rebate and a certain number of free weeks, as

shown in Table 7.2.

Table 7.1 Matrix A

Bronze

Silver

Table 7.2 Matrix B

Gold

Rebate

Free Weeks

Homes

40

20

25

Bronze

\$15

2

10

15

45

Silver

\$25

4

Gold

\$35

6

To compute the amount of rebate money the cable company paid to households

for the week, we would take the first row (R1) in Table 7.1 and multiply by the

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655

corresponding entries (bronze with bronze, silver with silver, and so on) in the first

column (C1) of Table 7.2 and add these products. In matrix form, we have

15

#

3 40 20 25 4 £ 25 § ϭ 40 # 15 ϩ 20 # 25 ϩ 25 # 35 ϭ \$1975. Using R1 of Table 7.1

35

with C2 from Table 7.2 gives the number of free weeks awarded to homes:

2

#

3 40 20 254 £ 4 § ϭ 40 # 2 ϩ 20 # 4 ϩ 25 # 6 ϭ 310. Using the second row (R2) of

6

Table 7.1 with the two columns from Table 7.2 will give the amount of rebate money

and the number of free weeks, respectively, awarded to business customers. When all

computations are complete, the result is a product matrix P with order 2 ϫ 2. This is

because the product of R1 from matrix A, with C1 from matrix B, gives the entry in

15 2

40 20 25 #

1975 310

d £ 25 4 § ϭ c

d.

position P11 of the product matrix: c

10 15 45

2100 350

35 6

#

Likewise, the product R1 C2 will give entry P12 (310), the product of R2 with C1 will

give P21 (2100), and so on. This “row ϫ column” multiplication can be generalized,

and leads to the following. Given m ϫ n matrix A and s ϫ t matrix B,

A

1m ϫ n2

c

B

1s ϫ t 2

A

1m ϫ n2

c

c

matrix multiplication is

possible only when

nϭs

B

1s ϫ t 2

c

result will be an

m ϫ t matrix

In more formal terms, we have the following definition of matrix multiplication.

Matrix Multiplication

Given the m ϫ n matrix A ϭ 3aij 4 and the s ϫ t matrix B ϭ 3bij 4. If n ϭ s, then

matrix multiplication is possible and the product AB is an m ϫ t matrix P ϭ 3 pij 4,

where pij is product of the ith row of A with the jth column of B.

In less formal terms, matrix multiplication involves multiplying the row entries of

the first matrix with the corresponding column entries of the second, and adding them

together. In Example 6, two of the matrix products [parts (a) and (b)] are shown in full

detail, with the first entry of the product matrix color-coded.

EXAMPLE 6

Multiplying Matrices

Given the matrices A through E shown here, compute the following products:

a. AB

Ϫ2

Aϭ c

3

b. CD

1

d

4

4

Bϭ c

6

3

d

1

c. DC

Ϫ2 1

Cϭ £ 1

0

4

1

d. AE

3

2 §

Ϫ1

2

D ϭ £4

0

e. EA

5

1

Ϫ1

1 §

3

Ϫ2

Ϫ2

Eϭ £ 3

1

Ϫ1

0 §

2

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CHAPTER 7 Matrices and Matrix Applications

Solution

Ϫ2

1 4 3

d ϭ c

dc

36

4 6 1

1Ϫ22142 ϩ 112162

Computation: c

132142 ϩ 142 162

a. AB ϭ c

Ϫ2

3

A

B

(2 ϫ 2) (2 ϫ 2)

Ϫ5

d

13

1Ϫ22 132 ϩ 112 112

d

132132 ϩ 142112

c

0 Ϫ2 Ϫ7

Ϫ2 1 3

2 5

1

b. CD ϭ £ 1 0 2 § £ 4 Ϫ1 1 § ϭ £ 2 11 Ϫ3 §

12 16 7

4 1 Ϫ1 0 3 Ϫ2

1Ϫ22122 ϩ 112142 ϩ (3)(0)

Computation: £ 112122 ϩ 102 142 ϩ 122(0)

142 122 ϩ 112 142 ϩ 1Ϫ12(0)

Ϫ2

1

d£ 3

4

1

Ϫ2

e. EA ϭ £ 3

1

Ϫ1

Ϫ2

0 §c

3

2

c

C

D

(3 ϫ 3) (3 ϫ 3)

C

D

(3 ϫ 3) (3 ϫ 3)

c

c

c

c

result will be a

3 ϫ 3 matrix

1Ϫ22 112 ϩ 112112 ϩ (3)1Ϫ22

112112 ϩ 102112 ϩ 1221Ϫ22 §

142112 ϩ 112112 ϩ 1Ϫ121Ϫ22

D

(3 ϫ 3)

c

C

(3 ϫ 3)

D

(3 ϫ 3)

c

c

multiplication is possible

since 3 ϭ 3

A

(2 ϫ 2)

Ϫ1

0 §

2

c

result will be a

2 ϫ 2 matrix

1Ϫ22152 ϩ 112 1Ϫ12 ϩ (3)132

112152 ϩ 1021Ϫ12 ϩ 122132

142 152 ϩ 1121Ϫ12 ϩ 1Ϫ12132

1

1

d ϭ £ Ϫ6

4

4

c

multiplication is possible

since 2 ϭ 2

multiplication is possible

since 3 ϭ 3

2 5

1

Ϫ2 1 3

5

3 15

c. DC ϭ £ 4 Ϫ1 1 § £ 1 0 2 § ϭ £Ϫ5 5 9 §

0 3 Ϫ2

4 1 Ϫ1

Ϫ5 Ϫ2 8

Ϫ2

d. AE ϭ c

3

A

B

(2 ϫ 2) (2 ϫ 2)

c

C

(3 ϫ 3)

c

result will be a

3 ϫ 3 matrix

E

(3 ϫ 2)

c

multiplication is not possible since 2

E

(3 ϫ 2)

Ϫ6

3 §

9

c

A

(2 ϫ 2)

E

(3 ϫ 2)

c

3

A

(2 ϫ 2)

c

multiplication is possible

since 2 ϭ 2

c

result will be a

3 ϫ 2 matrix

Now try Exercises 29 through 40

Example 6 shows that in general, matrix multiplication is not commutative. Parts

(b) and (c) show CD DC since we get different results, and parts (d) and (e) show

AE EA, since AE is not defined while EA is.

As with the addition and subtraction of matrices, matrix multiplication becomes

cumbersome and time consuming for larger matrices, and we will often turn to the

technology available in such cases.

EXAMPLE 7

Using Technology for Matrix Operations

Use a calculator to compute the product AB.

Solution

2

Ϫ1

Aϭ ≥

6

3

Carefully enter matrices A and B into the calculator,

then press 2nd MODE (QUIT) to get to the home

screen. Use [A][B] , and the calculator finds the

product shown in the figure. Just for “fun,” we’ll

Ϫ3

5

0

2

0

1

2

4

¥ B ϭ £ 0.5

2

Ϫ2

Ϫ1

A

(4 ϫ 3)

Ϫ0.7

3.2

3

4

B

(3 ϫ 3)

A

(4 ϫ 3)

c

c

1

Ϫ3 §

4

B

(3 ϫ 3)

ENTER

c

multiplication is possible

since 3 ϭ 3

c

result will be a

4 ϫ 3 matrix

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Section 7.2 The Algebra of Matrices

double check the first entry of the product matrix:

(2)(1/2) ϩ (Ϫ3)(0.5) ϩ (0)(Ϫ2) 0.5.

2

1

AB

6

3

3

5

0

2

0

1

2

4

Ơ Ê 0.5

2

2

1

0.7

3.2

3

4

1

3 Đ

4

Now try Exercises 41 through 52

Properties of Matrix Multiplication

Earlier, Example 6 demonstrated that matrix multiplication is not commutative. Here

is a group of properties that do hold for matrices. You are asked to check these properties in the Exercise Set using various matrices. See Exercises 53 through 56.

Properties of Matrix Multiplication

Given matrices A, B, and C for which the products are defined:

I. A1BC2 ϭ 1AB2C

II. A1B ϩ C2 ϭ AB ϩ AC

III. 1B ϩ C2A ϭ BA ϩ CA

IV. k1A ϩ B2 ϭ kA ϩ kB

S matrix multiplication is associative

S matrix multiplication is distributive from the left

S matrix multiplication is distributive from the right

S a constant k can be distributed over addition

We close this section with an application of matrix multiplication. There are many

other interesting applications in the Exercise Set.

EXAMPLE 8

Using Matrix Multiplication to Track Volunteer Enlistments

In a certain country, the number of males and females that will join the military depends on their age.

This information is stored in matrix A (Table 7.3). The likelihood a volunteer will join a particular

branch of the military also depends on their age, with this information stored in matrix B (Table 7.4).

(a) Compute the product P ϭ AB and discuss/interpret what is indicated by the entries P11, P13, and

P24 of the product matrix. (b) How many males are expected to join the Navy this year?

Table 7.4 Matrix B

Table 7.3 Matrix A

A

Solution

B

Age Group

Likelihood of Joining

Sex

18–19

20–21

22–23

Age Group

Army

Navy

Air Force

Marines

Female

1000

1500

500

18–19

0.42

0.28

0.17

0.13

Male

2500

3000

2000

20–21

0.38

0.26

0.27

0.09

22–23

0.33

0.25

0.35

0.07

a. Matrix A has order 2 ϫ 3 and matrix B has order 3 ϫ 4. The product matrix P can be found

and is a 2 ϫ 4 matrix. Carefully enter the matrices in your calculator. Figure 7.12 shows the

entries of matrix B. Using 3A 4 3 B4 , the calculator finds the product matrix shown in

Figure 7.13. Pressing the right arrow shows the complete product matrix is

ENTER

Pϭ c

1155

2850

795

1980

750

1935

300

d.

735

The entry P11 is the product of R1 from A and C1 from B, and indicates that for the year, 1155

females are projected to join the Army. In like manner, entry P13 shows that 750 females are

projected to join the Air Force. Entry P24 indicates that 735 males are projected to join the

Marines.

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CHAPTER 7 Matrices and Matrix Applications

Figure 7.13

Figure 7.12

b. The product R2 (males) # C2 (Navy) gives P22 ϭ 1980, meaning 1980 males are expected to

join the Navy.

Now try Exercise 59 through 66

C. You’ve just seen how

we can compute the product

of two matrices

7.2 EXERCISES

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

1. Two matrices are equal if they are like size and the

corresponding entries are equal. In symbols, A ϭ B

if

.

ϭ

2. The sum of two matrices (of like size) is found by

adding the corresponding entries. In symbols,

AϩBϭ

.

3. The product of a constant times a matrix is called

multiplication.

4. The size of a matrix is also referred to as its

1 2 3

d is

The order of A ϭ c

.

4 5 6

5. Give two reasons why matrix multiplication is

generally not commutative. Include several

examples using matrices of various sizes.

6. Discuss the conditions under which matrix

multiplication is defined. Include several examples

using matrices of various sizes.

.

State the order of each matrix and name the entries in

positions a12 and a23 if they exist. Then name the

position aij of the “5” in each.

1

7. c

5

2

9. c

0

Ϫ2

11. £ 0

5

Ϫ3

5

1

8

Ϫ1

13. c

19

8. £ Ϫ11 §

5

Ϫ3

d

Ϫ7

0.5

d

6

Ϫ7

4

2

10. £ Ϫ0.1

0.3

89

12. £ 13

2

Determine if the following statements are true, false, or

conditional. If false, explain why. If conditional, find values

of a, b, c, p, q, and r that will make the statement true.

3

2

14. ≥

Ϫ1

2

0.4

Ϫ3

55

8

1

34

5

1

21

0

14

132

11

116

Ϫ2

15. £ 2b

0

Ϫ7

5

Ϫ2

5

3

Ϫ5

Ϫ9

2p ϩ 1

16. Ê 1

q5

18

1

d c

164

4

2

4 12

13

10

1.5

Ơ c

1

0.5

3

1.4

0.4

a

c

4 Đ £6

3c

0

Ϫ5

12

9

3

Ϫ5

Ϫ3b

9

7

0 § ϭ £ 1

Ϫ2r

Ϫ2

2 12

d

8

1.3

d

0.3

Ϫ4

Ϫa §

Ϫ6

Ϫ5

3r

3p

2Ϫq

0 §

Ϫ8

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College Algebra G&M—

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For matrices A through J as given, perform the

indicated operation(s), if possible. If an operation

cannot be completed, state why. Use a calculator only

for those exercises designated by an icon.

Aϭ c

2

5

1

E ϭ £0

4

0.5

5 §

3

Ϫ2

Ϫ1

3

Ϫ1

Gϭ £ 0

Ϫ4

1

2

1

4

1

8

2

Bϭ £ 1 §

Ϫ3

3

d

8

2

C ϭ Ê 0.2

1

I

0

2 Đ

6

2

1

3

3

8

3

2

3

4

1

D Ê0

0

0

2 Đ

6

1

4

5

8

5

2

F c

H c

0

1

0

6

12

8

5

9

d

6

3

d

2

7

32

J

5

16

A c

5

3

5

8

Ơ

3

16

0

G

1

2

1

4

46. BH

47. DG

48. GD

2

22. A Ϫ J

23. G ϩ I

24. I Ϫ G

25. 3H Ϫ 2A

26. 2E ϩ 3G

32. HA

33. CB

34. FH

35. HF

36. EB

37. H

2

39. FE

38. F2

40. EF

1

4

Ϫ3

1

19

Hϭ ≥

8

1

1

19

16

3

4

3

8

11

16

45. HB

21. H ϩ J

0

1

0

12 Ϫ8 32

2 § Fϭ c

d

4

8 16

Ϫ6

44. GE

51. FG

31. AH

0

1

0

43. EG

20. G ϩ D

30. DE

1

D ϭ £0

0

0

d

1

42. HA

19. F ϩ H

29. ED

1

0

41. AH

49. C

2

28. F Ϫ F

3

Bϭ c

13

3 §

2 13

Ϫ2

Ϫ1

3

1

E ϭ £0

4

18. E ϩ G

1

E Ϫ 3D

2

4

d

9

13

Cϭ £ 2

13

0

1

Ϫ3

0

For matrices A through H as given, use a calculator to

perform the indicated operation(s), if possible. If an

operation cannot be completed, state why.

17. A ϩ H

27.

659

Section 7.2 The Algebra of Matrices

4

57

¥

5

57

50. E2

52. AF

For Exercises 53 through 56, use a calculator and

matrices A, B, and C to verify each statement.

Ϫ1

Aϭ £ 2

4

3

7

0

45

C ϭ £ Ϫ6

21

Ϫ1

10

Ϫ28

5

Ϫ1 §

6

0.3

B ϭ £ Ϫ2.5

1

Ϫ0.4

2

Ϫ0.5

1.2

0.9 §

0.2

3

Ϫ15 §

36

53. Matrix multiplication is not generally commutative:

(a) AB BA, (b) AC CA, and (c) BC CB.

54. Matrix multiplication is distributive from the left:

A1B ϩ C2 ϭ AB ϩ AC.

55. Matrix multiplication is distributive from the right:

1B ϩ C2A ϭ BA ϩ CA.

56. Matrix multiplication is associative:

1AB2C ϭ A1BC2.

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c

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CHAPTER 7 Matrices and Matrix Applications

WORKING WITH FORMULAS

2

W

2

d

0

#

c

L

Perimeter

d ‫ ؍‬c

d

W

Area

The perimeter and area of a rectangle can be simultaneously calculated using the matrix formula shown, where L

represents the length and W represents the width of the rectangle. Use the matrix formula and your calculator to find

the perimeter and area of the rectangles shown, then check the results using P ϭ 2L ϩ 2W and A ϭ LW.

57.

6.374 cm

4.35 cm

58.

5.02 cm

3.75 cm

APPLICATIONS

59. Custom T’s designs and sells specialty T-shirts and

sweatshirts, with factories in Verdi and Minsk. The

company offers this apparel in three quality levels:

standard, deluxe, and premium. Last fall the Verdi

plant produced 3820 standard, 2460 deluxe, and

1540 premium T-shirts, along with 1960 standard,

1240 deluxe, and 920 premium sweatshirts. The

Minsk plant produced 4220 standard, 2960 deluxe,

and 1640 premium T-shirts, along with 2960

standard, 3240 deluxe, and 820 premium

sweatshirts in the same time period.

a. Write a 3 ϫ 2 “production matrix” for each

plant 3 V S Verdi, M S Minsk], with a T-shirt

column, a sweatshirt column, and three rows

showing how many of the different types of

apparel were manufactured.

b. Use the matrices from part (a) to determine

how many more or fewer articles of clothing

were produced by Minsk than Verdi.

c. Use scalar multiplication to find how many

shirts of each type will be made at Verdi and

Minsk next fall, if each is expecting a 4%

d. Write a matrix that shows Custom T’s total

production next fall (from both plants), for

each type of apparel.

60. Terry’s Tire Store sells automobile and truck tires

through three retail outlets. Sales at the Cahokia store

for the months of January, February, and March

broke down as follows: 350, 420, and 530 auto tires

and 220, 180, and 140 truck tires. The Shady Oak

branch sold 430, 560, and 690 auto tires and 280,

320, and 220 truck tires during the same 3 months.

Sales figures for the downtown store were 864, 980,

and 1236 auto tires and 535, 542, and 332 truck tires.

a. Write a 2 ϫ 3 “sales matrix” for each store

3 C S Cahokia, S S Shady Oak, D S

Downtown], with January, February, and

March columns, and two rows showing the

sales of auto and truck tires respectively.

b. Use the matrices from part (a) to determine

how many more or fewer tires of each type the

downtown store sold (each month) over the

other two stores combined.

c. Market trends indicate that for the same three

months in the following year, the Cahokia

store will likely experience a 10% increase in

sales, the Shady Oak store a 3% decrease, with

sales at the downtown store remaining level

(no change). Write a matrix that shows the

combined monthly sales from all three stores

next year, for each type of tire.

61. Home improvements: Dream-Makers Home

Improvements specializes in replacement windows,

replacement doors, and new siding. During the

peak season, the number of contracts that came

from various parts of the city (North, South, East,

and West) are shown in matrix C. The average

profit per contract is shown in matrix P. Compute

the product PC and discuss what each entry of the

product matrix represents.

Windows

Doors

Siding

Windows

3 1500

N S E W

9 6 5 4

£7 5 7 6§ ϭ C

2 3 5 2

Doors Siding

500

2500 4 ϭ P

62. Classical music: Station 90.7 — The Home of

Classical Music — is having their annual fund

drive. Being a loyal listener, Mitchell decides that

for the next 3 days he will donate money according

to his favorite composers, by the number of times

their music comes on the air: \$3 for every piece by

Mozart (M), \$2.50 for every piece by Beethoven

(B), and \$2 for every piece by Vivaldi (V).

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College Algebra G&M—

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Section 7.2 The Algebra of Matrices

This information is displayed in matrix D. The

number of pieces he heard from each composer is

displayed in matrix C. Compute the product DC

and discuss what each entry of the product matrix

represents.

Mon. Tue. Wed.

M 4

3

5

B £3

2

4§ ϭ C

V 2

3

3

M B V

3 3 2.5 2 4 ϭ D

63. Pizza and salad: The science department and math

department of a local college are at a pre-semester

retreat, and decide to have pizza, salads, and soft

drinks for lunch. The quantity of food ordered by

each department is shown in matrix Q. The cost of

the food item at each restaurant is shown in matrix

C using the published prices from three popular

restaurants: Pizza Home (PH), Papa Jeff’s (PJ), and

Dynamos (D).

a. What is the total cost to the math department if

the food is ordered from Pizza Home?

b. What is the total cost to the science department

if the food is ordered from Papa Jeff’s?

c. Compute the product QC and discuss the

meaning of each entry in the product matrix.

Pizza

Science 8

c

Math 10

PH

Pizza

8

Drink 0.90

12

20

d ϭQ

8

18

PJ

D

7.5

10

1.75

2 § ϭC

1

0.75

64. Manufacturing pool tables: Cue Ball

Incorporated makes three types of pool tables, for

homes, commercial use, and professional use. The

amount of time required to pack, load, and install

each is summarized in matrix T, with all times in

hours. The cost of these components in dollars per

hour, is summarized in matrix C for two of its

warehouses, one on the west coast and the other in

the midwest.

a. What is the cost to package, load, and install a

commercial pool table from the coastal

warehouse?

b. What is the cost to package, load, and install a

commercial pool table from the warehouse in

the midwest?

c. Compute the product TC and discuss the

meaning of each entry in the product matrix.

661

Home

1

0.2 1.5

Comm £ 1.5

0.5 2.2 § ϭ T

Prof 1.75 0.75 2.5

Coast Midwest

Pack 10

8

10.5 § ϭ C

Install 13.5 12.5

65. Joining a club: Each school year, among the

students planning to join a club, the likelihood a

student joins a particular club depends on their

class standing. This information is stored in matrix

C. The number of males and females from each

class that are projected to join a club each year is

stored in matrix J. Compute the product JC and use

the result to answer the following:

a. Approximately how many females joined the

chess club?

b. Approximately how many males joined the

writing club?

c. What does the entry p13 of the product matrix

tells us?

Fresh

Female 25

c

Male 22

Spanish

Fresh 0.6

Soph £ 0.5

Junior 0.4

Soph Junior

18

21

d ϭJ

19

18

Chess

0.1

0.2

0.2

Writing

0.3

0.3 § ϭ C

0.4

66. Designer shirts: The SweatShirt Shoppe sells

three types of designs on its products: stenciled (S),

embossed (E), and applique (A). The quantity of

each size sold is shown in matrix Q. The retail

price of each sweatshirt depends on its size and

whether it was finished by hand or machine. Retail

prices are shown in matrix C. Assuming all stock is

sold, compute the product QC and use the result to

a. How much revenue was generated by the large

sweatshirts?

b. How much revenue was generated by the

extra-large sweatshirts?

c. What does the entry p11 of the product matrix

QC tell us?

S

med 30

large £ 60

x-large 50

E

30

50

40

A

15

20 § ϭ Q

30

Hand

S 40

E £ 60

A 90

Machine

25

40 § ϭ C

60

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EXTENDING THE CONCEPT

67. For the matrix A shown, use your calculator to

compute A2, A3, A4, and A5. Do you notice a

pattern? Try to write a “matrix formula” for An,

where n is a positive integer, then use your formula

to find A6. Check results using a calculator.

1 0 1

A ϭ £1 1 1§

1 0 1

7–26

CHAPTER 7 Matrices and Matrix Applications

68. The matrix M ϭ c

2

1

d has some very

Ϫ3 Ϫ2

interesting properties. Compute the powers M 2, M 3,

M 4, and M 5, then discuss what you find. Try to

find/create another 2 ϫ 2 matrix that has similar

properties.

69. (6.2) Solve the system using elimination.

x ϩ 2y Ϫ z ϭ 3

• Ϫ2x Ϫ y ϩ 3z ϭ Ϫ5

5x ϩ 3y Ϫ 2z ϭ 2

log2 21

70. (2.1/4.6) Given f(x) ϭ x ϩ 10x Ϫ 9, solve

f 1x2 Ն 0.

4

2

MID-CHAPTER

CHECK

MID-CHAPTER

Ϫ2

2. B ϭ c

4

1.1

0.1

0.4

1

1

2

3

4

0

0.2

Ϫ0.9 §

0.8

5

d

Ϫ3

Write each system in matrix form and solve using row

operations to triangularize the matrix. If the system is

linearly dependent, write the solution using a parameter.

3. e

Ϫx ϩ y Ϫ 5z ϭ 23

4. • 2x ϩ 4y Ϫ z ϭ 9

3x Ϫ 5y ϩ z ϭ 1

2x ϩ 3y ϭ Ϫ5

Ϫ5x Ϫ 4y ϭ 2

x ϩ y Ϫ 3z ϭ Ϫ11

5. • 4x Ϫ y Ϫ 2z ϭ Ϫ4

3x Ϫ 2y ϩ z ϭ 7

6. For matrices A and B given, compute:

Aϭ c

a. A Ϫ B

Ϫ3

5

Ϫ2

10

d Bϭ c

4

Ϫ30

b.

2

B

5

72. (4.1) Find the quotient using synthetic division,

then check using multiplication.

x3 Ϫ 9x ϩ 10

xϪ2

CHECK

State the size of each matrix and identify the entry in

second row, third column.

0.4

1. A ϭ £ Ϫ0.2

0.7

71. (5.4) Evaluate using the change-of-base formula,

then check using exponentiation.

7. For matrices C and D given, use a calculator to find:

Ϫ0.2

C ϭ £ 0.4

0.1

c. 5A ϩ B

0.2

5

0 § D ϭ £ Ϫ2.5

Ϫ0.1

10

2.5

0

2.5

10

Ϫ5 §

10

1

a. C ϩ D

b. Ϫ0.6D

c. CD

5

8. For the matrices A, B, C, and D given, compute the

products indicated (if possible):

4

Aϭ c

0

4

Cϭ c

Ϫ1

a. AC

15

d

Ϫ5

0

0.8

Ϫ0.2

Ϫ8

0

Ϫ2

1 §

7

6

B ϭ £0

4

Ϫ1

d

Ϫ5

Ϫ3

d

1

b. Ϫ2CD

2

D ϭ £ Ϫ1

1

c. BA

0

Ϫ3

5

Ϫ6

0 §

Ϫ4

d. CBϪ 4A

9. Create a system of equations to model this exercise,

then write the system in matrix form and solve. The

campus bookstore offers both new and used texts to

students. In a recent biology class with 24 students,

14 bought used texts and 10 bought new texts, with

the class as a whole paying \$2370. Of the 6 premed

students in class, 2 bought used texts, and 4 bought

new texts, with the group paying a total of \$660.

How much does a used text cost? How much does a

new text cost?

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Section 7.3 Solving Linear Systems Using Matrix Equations

10. Table A shown gives the number and type of extended warranties sold to individual car owners and to business

fleets. Table B shows the promotions offered to those making the purchase. Write the entries of each table in

matrix form and compute the product matrix P ϭ AB, then state what each entry of the product matrix represents.

Table B

Table A

Rebate

Free AAA

Membership

\$50

1 yr

100,000 mi

\$75

2 yr

120,000 mi

\$100

3 yr

Extended

Warranties

80,000

mi

100,000

mi

120,000

mi

Individuals

30

25

10

80,000 mi

20

12

5

Promotions

REINFORCING BASIC CONCEPTS

More on Matrix Multiplication

To help understand and master the concept of matrix multiplication, it helps to take a closer

look at the entries of the product matrix. Recall for the product AB ϭ P, the entry p11 in the

product matrix is the result of multiplying the 1st row of A with the 1st column of B, the entry

p12 is the result of multiplying 1st row of A, with the 2nd column of B, and so on.

1

P ϭ £6

7

2

Ϫ2 §

7

Exercise 1: The product of the 3rd row of A with the 2nd column of B, gives what entry in P?

Exercise 2: The entry p13 is the result of what product? The entry p22 is the result of what product?

Exercise 3: If p33 is the last entry of the product matrix, what are the possible sizes of A and B?

Exercise 4: Of the eight matrices shown here, only two produce the product matrix P shown. Use the ideas

highlighted above to determine which two.

3

A ϭ £2

4

Ϫ1

1 §

1

1

Bϭ c

2

1

d

0

2

C ϭ £ Ϫ1

1

1

Eϭ c

3

2

d

Ϫ1

1

F ϭ £0

4

0

1

1

G ϭ £4§

6

7.3

0

Ϫ3

5

Ϫ6

0 §

Ϫ4

2

D ϭ £ Ϫ1

3

H ϭ 31

4

5

34

Solving Linear Systems Using Matrix Equations

LEARNING OBJECTIVES

In Section 7.3 you will see

how we can:

A. Recognize the identity

matrix for multiplication

B. Find the inverse of a

square matrix

C. Solve systems using

matrix equations

D. Use determinants to find

whether a matrix is

invertible

While using matrices and row operations offers a degree of efficiency in solving systems, we are still required to solve for each variable individually. Using matrix multiplication we can actually rewrite a given system as a single matrix equation, in which the

solutions are computed simultaneously. As with other kinds of equations, the use of

identities and inverses are involved, which we now develop in the context of matrices.

A. Multiplication and Identity Matrices

From the properties of real numbers, 1 is the identity for multiplication since

n # 1 ϭ 1 # n ϭ n. A similar identity exists for matrix multiplication. Consider the 2 ϫ 2

1 4

matrix A ϭ c

d . While matrix multiplication is not generally commutative,

Ϫ2 3

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B. Addition and Subtraction of Matrices

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